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Theorem hmphtop 17469
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmphtop  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )

Proof of Theorem hmphtop
StepHypRef Expression
1 df-hmph 17447 . . 3  |-  ~=  =  ( `'  Homeo  " ( _V  \  1o ) )
2 cnvimass 5033 . . . 4  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 17448 . . . . 5  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5343 . . . . 5  |-  (  Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 8 . . . 4  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3210 . . 3  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3208 . 2  |-  ~=  C_  ( Top  X.  Top )
87brel 4737 1  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   1oc1o 6472   Topctop 16631    Homeo chmeo 17444    ~= chmph 17445
This theorem is referenced by:  hmphtop1  17470  hmphtop2  17471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-hmeo 17446  df-hmph 17447
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